We report a first-principles investigation of magnetic excitations in a compensated half-metallic ferrimagnet using both the adiabatic Heisenberg model and the dynamic spin susceptibility of the electronic system. The combination of half-metallicity and spin compensation of inequivalent magnetic sublattices generates two acoustic magnon modes characterized by linear dispersions with equal spin wave velocities and an asymmetric Landau damping. The difference in the damping is the consequence of the half-metallicity leading to the gap in the energy of the up-to-down electronic spin-flip excitations, whereas down-to-up excitations are gapless. The engineering of the activation energy of the Stoner excitations opens an avenue for the chirality-selective manipulation of the lifetime of magnons.